van 't Hoff equation
/ vant-HOF /
Suppose you measure a reaction's equilibrium constant at several temperatures and notice it drifts — bigger when warm for one reaction, smaller when warm for another. Is there a rule behind that drift? The van 't Hoff equation is that rule. It links how fast the equilibrium constant changes with temperature to a single property of the reaction: its enthalpy change, the heat it gives off or takes in.
The equation states that the rate of change of the natural logarithm of K with temperature is proportional to the reaction enthalpy ΔH divided by the temperature squared (and the gas constant). A handy consequence is its straight-line form: plotting ln K against 1 over the absolute temperature gives a line whose slope is −ΔH/R. From a few measurements you can thus read off the reaction's enthalpy, and predict K at a new temperature.
Why it matters: it makes the qualitative version of Le Chatelier's temperature rule quantitative. For an exothermic reaction (ΔH negative), K falls as temperature rises; for an endothermic reaction (ΔH positive), K climbs. The honest caveats are that the simple form assumes ΔH stays roughly constant over the temperature range, which is only approximately true, and that it concerns the constant K, not the speed of the reaction.
Measure K for an endothermic reaction at 300 K, 350 K and 400 K, then plot ln K against 1/T. The points fall on a downward-sloping line; multiply the slope by −R and you recover the positive reaction enthalpy ΔH — all without ever using a calorimeter.
Slope of ln K versus 1/T gives −ΔH/R.
Do not confuse the van 't Hoff equation (how K depends on temperature) with the Arrhenius equation (how a rate constant depends on temperature). They look alike and both give straight lines, but one is about equilibrium and the other about speed.